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A quadratic function is a type of polynomial function where the highest exponent of the variable is 2. It is generally written in the form:

• \( f(x) = ax^2 + bx + c \)

Here, \( a, b, \) and \( c \) represent constants, and \( a \) cannot be zero. This function constructs a parabola on a graph, which can either open upwards or downwards depending on the sign of \( a \). If \( a > 0 \), the parabola opens up; if \( a < 0 \), it opens down.

The focal point of a parabola, known as the vertex, is particularly important. It denotes the minimum or maximum value of the function. This vertex can be found using the formula:

• \( h = \frac{-b}{2a} \)

Substituting \( h \) back, you get the \( y \)-coordinate of the vertex, allowing you to determine the vertex \( (h, k) \). In the function \( g(x) = x^2 + 4x + 7 \), \( a = 1 \) and \( b = 4 \). Hence, using the formula gives us a vertex at \((-2, 3)\), though the restriction in the domain limits the function's actual starting point.

The range of a function is the set of possible outputs or y-values that a function can produce. For quadratic functions, the range is largely influenced by the vertex and the direction in which the parabola opens.

In the function \( g(x) = x^2 + 4x + 7 \), the parabola opens upwards, as indicated by the positive coefficient of \( x^2 \). Therefore, the range begins from the lowest y-value, occurring at the smallest x-value within the domain, extending to positive infinity.

Given the domain is restricted to \([1, \infty)\), substituting \( x = 1 \) into the function gives the lowest point within the domain, \( g(1) = 12 \). Thus, the range is from 12 to infinity: \([12, \infty)\).

The domain encompasses all possible input values (x-values) for a function, whereas the range covers all potential output values (y-values).

When considering inverses, it's essential to understand that the range of a function becomes the domain of its inverse, and vice versa. For the function \( g(x) = x^2 + 4x + 7 \) with domain \([1, \infty)\) and range \([12, \infty)\), its inverse function \( g^{-1}(x) \) retains this relationship.

• The domain of \( g^{-1}(x) \) is \([12, \infty)\)
• The range of \( g^{-1}(x) \) is \([1, \infty)\).

Understanding these transformations is crucial for solving and working with inverse functions correctly.

The quadratic formula is a powerful tool used to solve quadratic equations, particularly when the factorization method is not feasible. It is derived from the quadratic function:

• \( ax^2 + bx + c = 0 \).

The formula provides the solutions for x:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

It yields two solutions, arising from the "plus" and "minus" component in \( \pm \), representing the roots of the equation.

In the case of finding an inverse function such as \( g^{-1}(x) \), the quadratic formula assists in expressing \( y \) in terms of \( x \) when given \( x = y^2 + 4y + 7 \). Applying the formula enables us to simplify this to:

• \( g^{-1}(x) = \frac{-4 \pm \sqrt{4x - 12}}{2} \)

This form reveals how outputs from the function \( g \) inversely map to inputs of \( g^{-1} \).